Nope, they are supposed to be emotionless rock boulders.
Seriously, guys.
Any psychologist can confirm that showing emotions is requires strength, and is not necessarily a weakness, considering the person being well tempered in it and true to themselves, and that it often takes courage.
It is okay for men to cry, and bravely so.
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It could be a sign of weakness, unmanliness and cowardice in certain situations, though, but that's not the kind of crying that I'm talking about and the kind of tears a godly man will have shed. I go with what the Bible says is okay:
Reactions to extreme and tragic situations, great spiritual distress, repentance:
Genesis 27:38 And Esau said unto his father, Hast thou but one blessing, my father? bless me, even me also, O my father. And Esau lifted up his voice, and wept.
Psalms 6:6 I am weary with my groaning; all the night make I my bed to swim; I water my couch with my tears.
2 Kings 20:5 Turn again, and tell Hezekiah the captain of my people, Thus saith the LORD, the God of David thy father, I have heard thy prayer, I have seen thy tears: behold, I will heal thee: on the third day thou shalt go up unto the house of the LORD.
Job 16:20 My friends scorn me: but mine eye poureth out tears unto God.
Psalms 56:8 Thou tellest my wanderings: put thou my tears into thy bottle: are they not in thy book?
David seems to have cried quite a lot!
Deep occasions of expressing love (or mourning), either towards family members, closest friends, or a spouse:
Genesis 29:11 And Jacob kissed Rachel, and lifted up his voice, and wept.
Genesis 33:4 And Esau ran to meet him, and embraced him, and fell on his neck, and kissed him: and they wept.
Genesis 37:35 And all his sons and all his daughters rose up to comfort him; but he refused to be comforted; and he said, For I will go down into the grave unto my son mourning. Thus his father wept for him.
Genesis 43:30 And Joseph made haste; for his bowels did yearn upon his brother: and he sought where to weep; and he entered into his chamber, and wept there.
1 Samuel 20:41 And as soon as the lad was gone, David arose out of a place toward the south, and fell on his face to the ground, and bowed himself three times: and they (David and Jonathan) kissed one another, and wept one with another, until David exceeded.